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Solve the system of equations.\newliney=x2−20x+37y = x^2 - 20x + 37\newliney=−34x−11y = -34x - 11\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)

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Solve a system of linear and quadratic equations: parabolasright chevron icon

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Q. Solve the system of equations.\newliney=x2−20x+37y = x^2 - 20x + 37\newliney=−34x−11y = -34x - 11\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)
  1. Set Equations Equal: We have the system of equations:\newliney=x2−20x+37y = x^2 - 20x + 37\newliney=−34x−11y = -34x - 11\newlineSet the two equations equal to each other to find the xx-values where they intersect.\newlinex2−20x+37=−34x−11x^2 - 20x + 37 = -34x - 11
  2. Rearrange to Standard Form: Rearrange the equation to get a standard form quadratic equation.\newlinex2−20x+37+34x+11=0x^2 - 20x + 37 + 34x + 11 = 0\newlinex2+14x+48=0x^2 + 14x + 48 = 0
  3. Factor Quadratic Equation: Factor the quadratic equation.\newlineIn quadratic equation ax2+bx+cax^2 + bx + c, the factors are of the form (x+m)(x+n)(x + m)(x + n), where bb is the sum and cc is the product of mm and nn respectively.\newlinex2+14x+48=0x^2 + 14x + 48 = 0\newline(x+6)(x+8)=0(x + 6)(x + 8) = 0
  4. Solve for x: Solve for x.\newlineSet each factor equal to zero and solve for x.\newline(x+6)=0(x + 6) = 0 or (x+8)=0(x + 8) = 0\newlinex=−6x = -6 or x=−8x = -8
  5. Find Corresponding y-Values: Find the corresponding y-values for each x-value by substituting back into one of the original equations. We can use y=−34x−11y = -34x - 11.\newlineFor x=−6x = -6:\newliney=−34(−6)−11y = -34(-6) - 11\newliney=204−11y = 204 - 11\newliney=193y = 193
  6. Find y-Value: Find the y-value for x=−8x = -8:y=−34(−8)−11y = -34(-8) - 11y=272−11y = 272 - 11y=261y = 261
  7. Write Coordinates: Write the coordinates in exact form.\newlineFirst Coordinate: (−6,193)(-6, 193)\newlineSecond Coordinate: (−8,261)(-8, 261)

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